YOUR STUDY SPACE · CBSE

Predicting What Comes Next: Exploring Sequences and Progressions — Class 9 Mathematics

Ganita Manjari · Chapter 8

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Chapter summary

This chapter studies sequences — ordered lists of numbers whose entries are called terms — and ways to describe them. Familiar examples include natural, odd, triangular and square numbers. A sequence can be given by an explicit rule, which computes the th term directly from its position, such as for the odd numbers; or by a recursive rule, which builds each term from earlier ones, such as the Virahanka-Fibonacci rule . An arithmetic progression (AP) has a constant common difference ; its th term is , and its points lie on a straight line. A geometric progression (GP) has a constant common ratio ; its th term is , and its points do not lie on a line. The sum of the first natural numbers, first written by Aryabhata, is , which is also the th triangular number. Fractals such as the Sierpinski triangle and square carpet generate geometric progressions in both their counts and areas.

Sequences and termsExplicit and recursive rulesArithmetic progressionsGeometric progressionsSum of natural numbers and fractals

Key ideas & terms

Sequence: An ordered list of numbers, each of which is called a term.
Explicit rule: A formula giving the th term directly from its position , e.g. .
Recursive rule: A rule that defines each term using one or more previous terms.
Arithmetic progression: A sequence with a constant common difference ; .
Geometric progression: A sequence with a constant common ratio ; .
Common difference / ratio: The fixed amount added in an AP, or the fixed factor multiplied in a GP.

Extra questions & answers

What is the th term of the odd-number sequence?

It is , the explicit rule for .

What is an arithmetic progression?

A sequence in which consecutive terms differ by a constant common difference .

Write the formula for the th term of an AP.

It is , where is the first term and the common difference.

Find the th term of the AP .

Here : .

What is the formula for the th term of a GP?

It is , where is the first term and the common ratio.

Is a GP, and what is its common ratio?

Yes; each term is times the previous one, so the common ratio is .

What is the sum of the first natural numbers?

It is , also equal to the th triangular number.

Compute .

Using with : .

What is the recursive rule of the Virahanka-Fibonacci sequence?

With , it is for .

How does a GP graph differ from an AP graph?

AP points lie on a straight line, while GP points curve and do not lie on a straight line.

Think and Reflect

Can you describe the pattern in each of the above sequences? Can you predict the next few numbers in these sequences?

The four sequences are (Natural numbers) with rule and common difference ; next terms are . (Odd numbers) with rule and common difference ; next terms are . (Triangular numbers) where and differences are ; next terms are . (Square numbers) where and is the sum of first odd numbers; next terms are .

Exercise:

Consider the sequence 1, 4, 7, 10, 13, … Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? (Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is 1 + 4 + 7 = 12, and so on.)

Given sequence is arithmetic with , and . Next four terms after are , , , . Let . Then , , , , , , , , , .

Exercise:

Can you write , , and for the sequence of triangular numbers?

Triangular numbers are . Hence , , , .

Think and Reflect

Can you think of any other kinds of sequences? List out five different types of sequences and discuss their properties with your friends.

Examples: 1) Constant sequence with . 2) Decreasing arithmetic progression with . 3) Geometric progression with . 4) Alternating sequence with . 5) Fibonacci-type recursive sequence where . Each has different growth: constant no change, arithmetic linear, geometric exponential, alternating periodic, recursive depends on previous terms.

Think and Reflect

Why is it useful to have an explicit formula for the term of a sequence?

An explicit formula lets us find any term directly from its position without knowing all previous terms, check whether a given number belongs to the sequence and find its position by solving , and predict very large terms like the 1000th term quickly.

Exercise:

Using the explicit rule , find the 53rd term, the 108th term, and the 1170th term of the odd number sequence.

. Then , , .

Think and Reflect

Can you find the rule describing the term of the sequence of square numbers? Here is the sequence of the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Do you see any pattern in this sequence? Can you think of a rule that can predict the next few prime numbers?

Square numbers: so . For primes there is no simple arithmetic or geometric pattern and no elementary explicit formula that generates only primes. Differences are irregular. The next primes after are but they must be found by testing divisibility, not by a fixed difference or ratio. This shows not every sequence has a simple explicit rule.

Exercise:

Consider the expression . (i) Find its first, second, third, 12th, 18th and 50th terms.

, , , , , .

Consider the expression . (ii) Which term of the sequence is 332?

Solve giving so . Hence is the term.

Consider the expression . (iii) Is 557 a term of this sequence? Why or why not?

Solve giving so , which is a natural number. Hence is the term, so yes it is a term.

Exercise Set 8.1

1. Find the first five terms of the sequence in which the term is given by (i) for .

, , , , .

1. Find the first five terms of the sequence in which the term is given by (ii) for .

, , , , .

1. Find the first five terms of the sequence in which the term is given by (iii) for .

, , , , .

2. Find the 10th and 15th terms of the sequence for .

, .

3. Determine whether 97 and 172 are terms of the sequence for .

gives integer, so is the term. gives integer, so is the term. Both are terms.

4. Which term of the sequence for is 607?

gives so . The term is .

5. A sequence is given by the recursive rule , for . Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?

, , , , . Explicitly . Solve gives , so is the term.

6. Let , , , and for . Find , , , , and .

, , , , .

Think and Reflect

Can you predict the number of squares in Stages 5 and 6 of the sequence? In Stages 10, 11 and 12? In Stage 20? At any stage?

For the growing pattern of Fig 8.3, . Hence , , , , , . In general stage has tiny squares.

Think and Reflect

Consider all the sequences we have discussed so far in this chapter. Which ones are arithmetic progressions and which ones are not? Can you justify your claim?

Arithmetic progressions have constant difference : Natural numbers , odd numbers , , are APs. Triangular numbers differences not constant, square numbers differences not constant, primes have no constant difference, so they are not APs.

Exercise:

Verify that the following sequences are arithmetic progressions and write their terms. What do you observe when you plot the ordered pairs emerging from them? (i) 2, 5, 8, 11, …

Differences are , constant, so AP with , , . Plotting gives points lying on the straight line .

Verify that the following sequences are arithmetic progressions and write their terms. What do you observe when you plot the ordered pairs emerging from them? (ii) –5, –1, 3, 7, …

Differences are , constant, so AP with , , . Plotting gives points on the straight line .

Exercise:

Using the formula , find the term of the following arithmetic progressions. (i)

, . So .

Using the formula , find the term of the following arithmetic progressions. (ii) 1.5, 3.5, 5.5, 7.5, …

, . So .

Exercise:

Find recursive rules for the APs in the previous exercises.

In general , for . So (i) for : , . (ii) for : , . (iii) for : , . (iv) for : , .

Think and Reflect

Can the same approach be used to find the sum of ?

Yes. Let and reverse it. Adding gives , so .

Think and Reflect

Can you use this formula to find , or ?

Using : , , .

Think and Reflect

Let us revisit the sequence of triangular numbers 1, 3, 6, 10, 15, … shown in Fig. 8.1. Note that the term of this sequence is the sum of the first natural numbers. Thus . Can you use this to find the 10th, 17th and 80th triangular numbers?

, , .

Exercise Set 8.2

1. Find the 10th and 26th terms of the AP: 3, 8, 13, 18, ….

, , . So , .

2. Which term of the AP : 21, 18, 15, … is – 81? Also, is 0 a term of this AP? Give reasons for your answer.

, , . Solve gives , so term is . Solve gives , integer, so is the term.

3. Find the term of the AP: 11, 8, 5, 2 … Write the recursive rule for this AP.

, , so . Recursive: , for .

4. An AP consists of 50 terms in which the 3rd term is 12 and the last term is 106. Find the 29th term. (Hint: If ‘a’ is the first term and ‘d’ the common difference, then we arrive at the equations and . Solve this pair of linear equations for ‘a’ and ‘d’.)

Solving and gives so , . Then .

5. How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?

2-digit multiples of 3 form AP with . Number satisfies giving . Sum .

6. Harish started work at an annual salary of ₹5,00,000 and received an increment of ₹20,000 each year. After how many years did his income reach ₹7,00,000?

Annual salaries form AP , , . Set gives so . After 10 yearly increments, in the 11th year his salary is ₹7,00,000.

7. A child arranges marbles in rows so that the first row has 1 marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all?

Total marbles.

Think and Reflect

Can you predict the number of squares in Stages 5 and 6 of the pattern? In Stages 10, 11 and 12? In Stage 20? At any stage? How is this different from the growing pattern in Fig. 8.3?

For Fig 8.6 the count is geometric: . So , , , , , . This grows exponentially by multiplying by each stage, whereas the Fig 8.3 pattern is arithmetic growing linearly by adding each stage.

Exercise:

Check whether the following sequences are geometric progressions and find their terms. (i) 2, 10, 50, 250, …

Ratios , constant, so GP with , , .

Check whether the following sequences are geometric progressions and find their terms. (ii)

Ratios , constant, so GP with , , .

Check whether the following sequences are geometric progressions and find their terms. (iii)

The ratios are , and — a constant ratio, so it is a GP with first term and common ratio . The term is .

Exercise:

Can you find a recursive rule for the formula that generates the geometric progression 3, 30, 300, 3000, … ?

and each term is times previous, so for .

Think and Reflect

Observe the Sierpiński triangle and try to answer the following questions (a) How many black triangles are there in Stages 0 to 3 of Fig. 8.7?

Stage 0: , Stage 1: , Stage 2: , Stage 3: black triangles.

(b) Can you predict the number of black triangles at Stages 4 and 5?

, .

(c) Can you find a rule for the number of black triangles at the stage?

for starting at . If numbering from , . Recursion: , .

(d) Suppose the area of the triangle (that is, the black region) in Stage 0 is 1 square unit. What is the area of the black region in Stages 1, 2 and 3? What will be the area of the black region in Stages 4 and 5? Find a rule for the area of the black region at the stage. What happens to this area as , the number of stages, goes on increasing?

Each stage keeps of previous area. So , , , , . In general . As , , so black area tends to zero while number of triangles blows up.

Exercise Set 8.3

1. Find the 12th term of a GP with common ratio 2, whose 8th term is 192.

Let GP be . gives . Then .

2. Find the 10th and terms of the GP: 5, 25, 125, … .

, , . So .

3. A sequence is given by the recursive rule , for . Which term of the sequence is 730?

Compute: , , , , , , . So is the term.

4. Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the term.

, , explicit . Solve gives so . Recursive: , for .

5. A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell. It continues bouncing in this way — each time rising to 60% of the previous height. (i) What height does the ball reach after the 5th bounce?

Heights form GP . metres (approx ). The successive bounce heights are m.

5. A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell. It continues bouncing in this way — each time rising to 60% of the previous height. (ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the 6th time?

Total = initial drop plus twice sum of first 5 bounce heights (up and down). Sum . Total distance metres .

6. Which term of the sequence is 128?

This is GP with , . . Set gives so and . The term is .

7. Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on. Look at Fig. 8.12 and try to answer the following questions. (i) How many red squares are there in Stages 0 to 3?

Stage 0: , Stage 1: , Stage 2: , Stage 3: red squares.

7. Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. (ii) Can you predict the number of red squares in Stages 4 and 5?

, .

7. Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. (iii) Can you find a rule for the number of red squares at the stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage.

Explicit: (with ). Recursive: , for . If indexing from , for the same picture shifted.

7. Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. (iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area of the red region in Stages 4 and 5? Find the explicit as well as the recursive formula for the area of the red region at the stage. What happens to this area as , the number of stages, goes on increasing?

Each stage retains of previous area. So , , , , . Explicit , recursive , . As , .

End-of-Chapter Exercises

1. Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.

Let and . Subtract gives so , . Then .

2. Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.

and gives so , . AP is with .

3. How many three-digit numbers are divisible by 7? (Hint: All three-digit numbers divisible by 7 form an AP. Find the smallest and largest such three-digit numbers.)

Smallest 3-digit multiple of 7 is , largest is . AP gives so . There are such numbers.

4. How many multiples of 4 lie between 10 and 250? (Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)

Multiples between 10 and 250 are . gives so . There are multiples.

5. Find a GP for which the sum of the first two terms is – 4 and the fifth term is 4 times the third term.

Let and giving so or . If , gives GP with . If , gives GP with .

6. Find all possible ways of expressing 100 as the sum of consecutive natural numbers.

We need with integers , . Solving gives viable : ; gives ; gives . Allowing the trivial single term, these are the representations with natural numbers.

7. The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the 2nd hour, 4th hour and hour?

GP with , . After hours . So after 2 hours , after 4 hours , after hours . If counting the initial as first term, the sequence is .

8. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.

gives , gives . Subtract gives so , . First three terms .

9. Find the smallest value of such that the sum of the first natural numbers is greater than 1,000.

Need . gives , gives . So smallest .

10. Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the term.

, , explicit . Solve gives so . Recursive: , .

11. The sum of the first three terms of a GP is and their product is –1. Find the common ratio and the terms.

Let terms be . Product so . Sum gives leading to so or . Terms are or (same set).

12. If the 4th, 10th and 16th terms of a GP are , and respectively, prove that , , are in GP.

Let GP be . Then , , . Then and , so , which means , hence are in GP with ratio .

13. The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.

Let terms . and . Solving gives or . If , gives ; if , gives . Both are same progression reversed.

14. Suppose , and for , . Find the values of , , …, . Can you find a simpler recursive formula for ? Can you give an explicit formula?

Compute: , , , , , , , . Observe for . Explicit: for (gives ).

15. Suppose , and for , . Find the values of , , …, . Do you recognise this sequence?

Compute: , , , , , , , . This is the Virahanka-Fibonacci sequence where each term is sum of previous two.